Re: Countable models of ZFC



Rupert wrote:
herbzet wrote:
Rupert wrote:
herbzet wrote:

[...]

ZFC is a first-order theory. A first-order theory talks
about its universe of discourse, presumably non-empty. The
universe of discourse consists of the things the quantifiers
of the theory range over.

A "model" of ZFC, in the looser, non-model-theoretic sense, is
any thing or things that the theorems of ZFC are true of. These
will be the things that quantifiers range over. These things that
are quantified over are conventionally referred to as "sets".

But is the universe of discourse a "set"?

No.

Might not some set be the domain of a model of ZFC?


Of course. I thought you were talking about the intended model.

Um, yes I was. There was some conceptual slippage between my
two posts. Actually, my views are evolving each day as I
think about the subject and read the postings.

Once you have the intended model, then the question could be asked
if any element of the intended model could itself serve as the
domain of another model of the formalism. It could also be asked
if the original intended model by any chance happened to be an
element of a larger model.

But it seems to me you've got to have that intended model first.
It appears kind of magical to me that you guys are making all
these assertions about the properties that models of ZFC do
or don't have, when no model of ZFC has been adduced in the
first place. :-\ I'm just assuming it's possible for you
to do this on a hypothetical basis.

[...]

Quite so. I am not criticizing your mathematics. I am exploiting
this forum to question the logical/epistomological/philosophical
underpinnings of your mathematics.


Well, perhaps you could tell me just once more exactly what the
philosophical issue is. Maybe you could start a new thread about it.

Well, briefly, you've got this formalism and one wants to know
what, if anything, it may be taken as referring to, if one considers
it, as is conventional, as a language. Since practically all of
mathematics can be proven in this formalism, one would like to
to know just what the hell are the referents of its terms.

It seems unsatisfyingly circular to present as a semantics
some objects constructed in the formalism itself, even on a
hypothetical basis. If we don't know what the primitive terms
are referring to in the first place, then what are these objects
constructed with them supposed to be?

As far as philosophy goes: do you understand the first-order language
of set theory or don't you? If you don't, then I guess you're going to
have a problem understanding mathematics. But most mathematicians
think they can understand the first-order language of set theory. They
feel that they know what they're talking about. They feel that they
can make sense of the idea of a sentence of the first-order language
of set theory being true, not just true in this model or that model of
the axioms. If you don't feel this idea has any meaning, fine, but
then this will be a problem with all of mathematics, not just the part
I'm talking about.

The problems come in with questions like, is the (generalized)
continuum hypothesis true? Do various large cardinals exist?
What could it mean that there are countable models of ZFC?


Well, elaborate on what the problems are.

Well, take GCH. Is it true or not? The axioms of ZFC don't decide
the issue, it's true in some models and not in others. If we
"understand the first-order language of set-theory" we should be
able to say which models are in accord with our understanding and
which are not, and so decide the issue.

This situation bears a resemblance to the situation in
geometry: which geometry is true, Euclidean or Riemann or
Lobachevsky or what?

Do we resolve the issue as we do in geometry and say with
a catholic unconcern that they are all equally true, or do
we reject the resemblance as misleading, and say that set
theory bears a special significance and that GCH is true
or false based on our "understanding" of what sets are
supposed to be?

--
hz
.

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